Notes on Hazard-Free Circuits

TL;DR

This paper characterizes hazard-free circuits, proves their equivalence to circuits producing all prime implicants and implicates, and demonstrates super-polynomial complexity gaps for certain functions, with implications for circuit design and cybersecurity.

Contribution

It provides a complete syntactic characterization of hazard-free circuits, extends classical results to general circuits, and establishes complexity bounds for specific Boolean functions.

Findings

Hazard-free circuits produce all prime implicants and implicates.

Optimal hazard-free circuits for monotone functions are monotone.

Certain simple functions have super-polynomial hazard-free complexity.

Abstract

The problem of constructing hazard-free Boolean circuits (those avoiding electronic glitches) dates back to the 1940s and is an important problem in circuit design and even in cybersecurity. We show that a DeMorgan circuit is hazard-free if and only if the circuit produces (purely syntactically) all prime implicants as well as all prime implicates of the Boolean function it computes. This extends to arbitrary DeMorgan circuits a classical result of Eichelberger [IBM J. Res. Develop., 9 (1965)] showing this property for special depth-two circuits. Via an amazingly simple proof, we also strengthen a recent result Ikenmeyer et al. [J. ACM, 66:4 (2019)]: not only the complexities of hazard-free and monotone circuits for monotone Boolean functions do coincide, but every optimal hazard-free circuit for a monotone Boolean function must be monotone. Then we show that hazard-free circuit complexity of a very simple (non-monotone) Boolean function is super-polynomially larger than its unrestricted circuit complexity. This function accepts a Boolean n x n matrix iff every row and every column has exactly one 1-entry. Finally, we show that every Boolean function of n variables can be computed by a hazard-free circuit of size O(2^n/n).